OPTIMAL RISKY PORTFOLIOS- ASSET ALLOCATIONS BKM Ch 7

ASSET ALLOCATION  Idea  from bank account to diversified portfolio  principles are the same for any number of stocks  Discussion  A. bonds and stocks  B. bills, bonds and stocks  C. any number of risky assets 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 2

DIVERSIFICATION AND PORTFOLIO RISK  Market risk  Systematic or nondiversifiable  Firm-specific risk  Diversifiable or nonsystematic 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 3

FIGURE 7.1 PORTFOLIO RISK AS A FUNCTION OF THE NUMBER OF STOCKS IN THE PORTFOLIO 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 4

FIGURE 7.2 PORTFOLIO DIVERSIFICATION 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 5

COVARIANCE AND CORRELATION  Portfolio risk depends on the correlation between the returns of the assets in the portfolio  Covariance and the correlation coefficient provide a measure of the way returns two assets vary 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 6

TWO-SECURITY PORTFOLIO: RETURN 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 7

= Variance of Security D = Variance of Security E = Covariance of returns for Security D and Security E TWO-SECURITY PORTFOLIO: RISK 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 8

TWO-SECURITY PORTFOLIO: RISK CONTINUED  Another way to express variance of the portfolio: 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 9

 D,E = Correlation coefficient of returns Cov(r D, r E ) =  DE  D  E  D = Standard deviation of returns for Security D  E = Standard deviation of returns for Security E COVARIANCE 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 10

Range of values for  1, >  >-1.0 If  = 1.0, the securities would be perfectly positively correlated If  = - 1.0, the securities would be perfectly negatively correlated CORRELATION COEFFICIENTS: POSSIBLE VALUES 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 11

TABLE 7.1 DESCRIPTIVE STATISTICS FOR TWO MUTUAL FUNDS 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 12

 2 p = w 1 2  w 2 2  w 1 w 2 Cov(r 1, r 2 ) + w 3 2  3 2 Cov(r 1, r 3 ) + 2w 1 w 3 Cov(r 2, r 3 )+ 2w 2 w 3 THREE-SECURITY PORTFOLIO 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 13

ASSET ALLOCATION  Portfolio of 2 risky assets (cont’d)  examples  BKM7 Tables 7.1 & 7.3  BKM7 Figs. 7.3 (return), 7.4 (risk) & 7.5 (trade-off)  portfolio opportunity set (BKM7 Fig. 7.5)  minimum variance portfolio  choose w D such that portfolio variance is lowest  optimization problem  minimum variance portfolio has less risk  than either component (i.e., asset) 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 14

TABLE 7.2 COMPUTATION OF PORTFOLIO VARIANCE FROM THE COVARIANCE MATRIX 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 15

TABLE 7.3 EXPECTED RETURN AND STANDARD DEVIATION WITH VARIOUS CORRELATION COEFFICIENTS 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 16

FIGURE 7.3 PORTFOLIO EXPECTED RETURN AS A FUNCTION OF INVESTMENT PROPORTIONS 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 17

FIGURE 7.4 PORTFOLIO STANDARD DEVIATION AS A FUNCTION OF INVESTMENT PROPORTIONS 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 18

MINIMUM VARIANCE PORTFOLIO AS DEPICTED IN FIGURE 7.4  Standard deviation is smaller than that of either of the individual component assets  Figure 7.3 and 7.4 combined demonstrate the relationship between portfolio risk 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 19

FIGURE 7.5 PORTFOLIO EXPECTED RETURN AS A FUNCTION OF STANDARD DEVIATION 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 20

 The relationship depends on the correlation coefficient  -1.0 <  < +1.0  The smaller the correlation, the greater the risk reduction potential  If  = +1.0, no risk reduction is possible CORRELATION EFFECTS 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 21

FIGURE 7.6 THE OPPORTUNITY SET OF THE DEBT AND EQUITY FUNDS AND TWO FEASIBLE CALS 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 22

THE SHARPE RATIO  Maximize the slope of the CAL for any possible portfolio, p  The objective function is the slope: 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 23

FIGURE 7.7 THE OPPORTUNITY SET OF THE DEBT AND EQUITY FUNDS WITH THE OPTIMAL CAL AND THE OPTIMAL RISKY PORTFOLIO 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 24

FIGURE 7.8 DETERMINATION OF THE OPTIMAL OVERALL PORTFOLIO 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 25

ASSET ALLOCATION  Finding the optimal risky portfolio: II. Formally  Intuitively  BKM7 Figs. 7.6 and 7.7  improve the reward-to-variability ratio  optimal risky portfolio  tangency point (Fig. 7.8)  Formally: 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 26

ASSET ALLOCATION 18  formally (continued) 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 27

ASSET ALLOCATION 19  Example (BKM7 Fig. 7.8)  1. plot D, E, riskless  2. compute optimal risky portfolio weights  w D = Num/Den = 0.4; w E = 1- w D = 0.6  3. given investor risk aversion (A=4), compute w *  bottom line: 25.61% in bills; 29.76% in bonds ( x 0.4); rest in stocks 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 28

FIGURE 7.9 THE PROPORTIONS OF THE OPTIMAL OVERALL PORTFOLIO 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 29

MARKOWITZ PORTFOLIO SELECTION MODEL  Security Selection  First step is to determine the risk-return opportunities available  All portfolios that lie on the minimum- variance frontier from the global minimum- variance portfolio and upward provide the best risk-return combinations 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 30

MARKOWITZ PORTFOLIO SELECTION MODEL  Combining many risky assets & T-Bills  basic idea remains unchanged  1. specify risk-return characteristics of securities  find the efficient frontier (Markowitz)  2. find the optimal risk portfolio  maximize reward-to-variability ratio  3. combine optimal risk portfolio & riskless asset  capital allocation 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 31

 finding the efficient frontier  definition  set of portfolios with highest return for given risk   minimum-variance frontier  take as given the risk-return characteristics of securities  estimated from historical data or forecasts  n securities  n return + n(n-1) var. & cov.  use an optimization program  to compute the efficient frontier (Markowitz)  subject to same constraints MARKOWITZ PORTFOLIO SELECTION MODEL 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 32

 Finding the efficient frontier (cont’d)  optimization constraints  portfolio weights sum up to 1  no short sales, dividend yield, asset restrictions, …  Individual assets vs. frontier portfolios  BKM7 Fig  short sales  not on the efficient frontier  no short sales  may be on the frontier  example: highest return asset MARKOWITZ PORTFOLIO SELECTION MODEL 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 33

FIGURE 7.10 THE MINIMUM-VARIANCE FRONTIER OF RISKY ASSETS 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 34

MARKOWITZ PORTFOLIO SELECTION MODEL CONTINUED  We now search for the CAL with the highest reward-to-variability ratio 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 35

FIGURE 7.11 THE EFFICIENT FRONTIER OF RISKY ASSETS WITH THE OPTIMAL CAL 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 36

MARKOWITZ PORTFOLIO SELECTION MODEL CONTINUED  Now the individual chooses the appropriate mix between the optimal risky portfolio P and T-bills as in Figure 7.8 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 37

FIGURE 7.12 THE EFFICIENT PORTFOLIO SET 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 38

CAPITAL ALLOCATION AND THE SEPARATION PROPERTY  The separation property tells us that the portfolio choice problem may be separated into two independent tasks  Determination of the optimal risky portfolio is purely technical  Allocation of the complete portfolio to T- bills versus the risky portfolio depends on personal preference 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 39

FIGURE 7.13 CAPITAL ALLOCATION LINES WITH VARIOUS PORTFOLIOS FROM THE EFFICIENT SET 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 40

THE POWER OF DIVERSIFICATION  Remember:  If we define the average variance and average covariance of the securities as:  We can then express portfolio variance as: 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 41

TABLE 7.4 RISK REDUCTION OF EQUALLY WEIGHTED PORTFOLIOS IN CORRELATED AND UNCORRELATED UNIVERSES 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 42

RISK POOLING, RISK SHARING AND RISK IN THE LONG RUN  Consider the following: 1 − p =.999 p =.001 Loss: payout = $100,000 No Loss: payout = 0 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 43

RISK POOLING AND THE INSURANCE PRINCIPLE  Consider the variance of the portfolio:  It seems that selling more policies causes risk to fall  Flaw is similar to the idea that long-term stock investment is less risky 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 44

RISK POOLING AND THE INSURANCE PRINCIPLE CONTINUED  When we combine n uncorrelated insurance policies each with an expected profit of $, both expected total profit and SD grow in direct proportion to n: 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 45

RISK SHARING  What does explain the insurance business?  Risk sharing or the distribution of a fixed amount of risk among many investors 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 46

AN ASSET ALLOCATION PROBLEM BAHATTIN BUYUKSAHIN, JHU, INVESTMENT1/16/

AN ASSET ALLOCATION PROBLEM 2  Perfect hedges (portfolio of 2 risky assets)  perfectly positively correlated risky assets  requires short sales  perfectly negatively correlated risky assets BAHATTIN BUYUKSAHIN, JHU, INVESTMENT1/16/

AN ASSET ALLOCATION PROBLEM 3 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT1/16/

CHAPTER 8 Index Models

FACTOR MODEL BAHATTIN BUYUKSAHIN, JHU, INVESTMENT  Idea  the same factor(s) drive all security returns  Implementation (simplify the estimation problem)  do not look for equilibrium relationship  between a security’s expected return  and risk or expected market returns  look for a statistical relationship  between realized stock return  and realized market return 1/16/

FACTOR MODEL 2 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT  Formally  stock return  = expected stock return  + unexpected impact of common (market) factors  + unexpected impact of firm-specific factors 1/16/

INDEX MODEL BAHATTIN BUYUKSAHIN, JHU, INVESTMENT  Factor model  problem  what is the factor?  Index Model  solution  market portfolio proxy  S&P 500, Value Line Index, etc. 1/16/

 Reduces the number of inputs for diversification  Easier for security analysts to specialize ADVANTAGES OF THE SINGLE INDEX MODEL 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 54

ß i = index of a securities’ particular return to the factor m = Unanticipated movement related to security returns e i = Assumption: a broad market index like the S&P 500 is the common factor. SINGLE FACTOR MODEL 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 55

SINGLE-INDEX MODEL  Regression Equation:  Expected return-beta relationship: 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 56

SINGLE-INDEX MODEL CONTINUED  Risk and covariance:  Total risk = Systematic risk + Firm-specific risk:  Covariance = product of betas x market index risk:  Correlation = product of correlations with the market index 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 57

INDEX MODEL AND DIVERSIFICATION  Portfolio’s variance:  Variance of the equally weighted portfolio of firm-specific components:  When n gets large, becomes negligible 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 58

FIGURE 8.1 THE VARIANCE OF AN EQUALLY WEIGHTED PORTFOLIO WITH RISK COEFFICIENT Β P IN THE SINGLE-FACTOR ECONOMY 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 59

FIGURE 8.2 EXCESS RETURNS ON HP AND S&P 500 APRIL 2001 – MARCH /16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 60

FIGURE 8.3 SCATTER DIAGRAM OF HP, THE S&P 500, AND THE SECURITY CHARACTERISTIC LINE (SCL) FOR HP 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 61

TABLE 8.1 EXCEL OUTPUT: REGRESSION STATISTICS FOR THE SCL OF HEWLETT- PACKARD 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 62

FIGURE 8.4 EXCESS RETURNS ON PORTFOLIO ASSETS 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 63

ALPHA AND SECURITY ANALYSIS  Macroeconomic analysis is used to estimate the risk premium and risk of the market index  Statistical analysis is used to estimate the beta coefficients of all securities and their residual variances, σ 2 ( e i )  Developed from security analysis 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 64

ALPHA AND SECURITY ANALYSIS CONTINUED  The market-driven expected return is conditional on information common to all securities  Security-specific expected return forecasts are derived from various security-valuation models  The alpha value distills the incremental risk premium attributable to private information  Helps determine whether security is a good or bad buy 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 65

SINGLE-INDEX MODEL INPUT LIST  Risk premium on the S&P 500 portfolio  Estimate of the SD of the S&P 500 portfolio  n sets of estimates of  Beta coefficient  Stock residual variances  Alpha values 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 66

OPTIMAL RISKY PORTFOLIO OF THE SINGLE- INDEX MODEL  Maximize the Sharpe ratio  Expected return, SD, and Sharpe ratio: 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 67

OPTIMAL RISKY PORTFOLIO OF THE SINGLE- INDEX MODEL CONTINUED  Combination of:  Active portfolio denoted by A  Market-index portfolio, the (n+1)th asset which we call the passive portfolio and denote by M  Modification of active portfolio position:  When 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 68

THE INFORMATION RATIO  The Sharpe ratio of an optimally constructed risky portfolio will exceed that of the index portfolio (the passive strategy): 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 69

FIGURE 8.5 EFFICIENT FRONTIERS WITH THE INDEX MODEL AND FULL-COVARIANCE MATRIX 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 70

TABLE 8.2 COMPARISON OF PORTFOLIOS FROM THE SINGLE-INDEX AND FULL- COVARIANCE MODELS 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 71

INDEX MODEL: INDUSTRY PRACTICES BAHATTIN BUYUKSAHIN, JHU, INVESTMENT  Beta books  Merrill Lynch  monthly, S&P 500  Value Line  weekly, NYSE  etc.  Idea  regression analysis 1/16/

INDEX MODEL: INDUSTRY PRACTICES 2 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT  Example (Merrill Lynch differences, Table 8.3)  total (not excess) returns  slopes are identical  smallness  percentage price changes  dividends?  S&P 500  adjusted beta  beta = (2/3) estimated beta + (1/3). 1  sampling errors, convergence of new firms  exploiting alphas (Treynor-Black) 1/16/

TABLE 8.3 MERRILL LYNCH, PIERCE, FENNER & SMITH, INC.: MARKET SENSITIVITY STATISTICS 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 74

TABLE 8.4 INDUSTRY BETAS AND ADJUSTMENT FACTORS 1/16/2010BAHATTIN BUYUKSAHIN, JHU, INVESTMENT 75

USING INDEX MODELS BAHATTIN BUYUKSAHIN, JHU, INVESTMENT1/16/

USING INDEX MODELS 2 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT1/16/

USING INDEX MODELS 3 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT1/16/

USING INDEX MODELS 4 BAHATTIN BUYUKSAHIN, JHU, INVESTMENT1/16/